Power Systems

Pavlos S. Georgilakis

Spotlight on Modern Transformer Design With 121 figures

123

Pavlos S. Georgilakis, Asst. Prof. Department of Production Engineering and Management Technical University of Crete University Campus 731 00 Chania Greece pgeorg@dpem.tuc.gr

ISSN 1612-1287 ISBN 978-1-84882-666-3 e-ISBN 978-1-84882-667-0 DOI 10.1007/978-1-84882-667-0 Springer Dordrecht Heidelberg London New York British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Library of Congress Control Number: 2009928676 © Springer-Verlag London Limited 2009 Apart from any fair dealing for the purposes of research or private study, or criticism or review, aspermitted under the Copyright, Designs and Patents Act 1988, this publication may only bereproduced, stored or transmitted, in any form or by any means, with the prior permission in writing ofthe publishers, or in the case of reprographic reproduction in accordance with the terms of licences issued by the Copyright Licensing Agency. Enquiries concerning reproduction outside those termsshould be sent to the publishers. The use of registered names, trademarks, etc. in this publication does not imply, even in the absence ofa specific statement, that such names are exempt from the relevant laws and regulations and thereforefree for general use. The publisher makes no representation, express or implied, with regard to the accuracy of theinformation contained in this book and cannot accept any legal responsibility or liability for any errorsor omissions that may be made. Cover design: deblik, Berlin, Germany Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

This book is dedicated to my family

Foreword

Power transformer advantages of high efficiency and reliability have certainly contributed to the domination of alternating current in power networks since the beginning of the last century. From early times, their design has been a major con-cern and has been the subject of extended research. The first efforts were based on conveniently adapted analytical solutions enabling one to optimize their construc-tion and to take advantage of the improvements in magnetic and electric material properties.

During recent decades the development of the philosophy of transformer design has been a logical extension of the use of computers and numerical tools enabling one to model accurately the geometrical complexities as well as the nonlinear ma-terial characteristics for problem analysis. In addition, optimization algorithms have been very successfully combined with numerical techniques to represent the electromagnetic and thermal phenomena developed in power transformers, result-ing in very powerful composite computational methodologies. In particular, artifi-cial intelligence algorithms incorporated in such techniques have dramatically en-hanced the speed and capability for achieving detailed optimum designs.

With this book Professor Pavlos Georgilakis contributes to the diffusion of composite numerical methodologies for power transformer design based on the combination of standard design techniques for transformers with advanced nu-merical methods such as the finite element method, and efficient optimization al-gorithms such as sequential quadratic programming, the branch-and-bound tech-nique, genetic algorithms, decision trees and artificial neural networks. The proposed approach to the subject creates a proper link between the various meth-odologies implemented and their particular contribution to this field. The impor-tant problem of transformer selection criteria is systematically treated by using total owning cost considerations and external environmental cost issues.

The author’s involvement in research both in the design office of a transformer construction company and the Technical University of Crete has given him a wide experience of the subject. His previous industry experience is reflected in the book by many references to actual practices. His academic background and the number of papers he has published in refereed journals ensure that a thorough theoretical treatment is given to important topics.

An important advantage of this work is that all methodologies presented are il-lustrated through detailed practical examples, concerning general power trans-former construction, including also the shell type core transformer case. The pro-posed examples cover all features of power transformer design and have been

viii Foreword

worked out in a rigorous and coherent manner. The introductory detailed presenta-tion of the fundamental topics and phenomena involved enables the implementa-tion of a textbook for teaching step by step the mysteries of transformer design both at undergraduate and postgraduate level. Certainly, it constitutes an excellent reference for researchers in the field, practicing electrical engineers and trans-former design office use.

Without any doubt, the book fills an important knowledge gap in our energy conservation challenges!

Athens, Greece John A. Tegopoulos February 2009 Life Fellow IEEE Professor Emeritus National Technical University of Athens

Preface

Many of the standard books on transformers are now over ten years old and some much older. Much has changed in the transformer industry since these books were written. Newer and better materials are now available for core and winding con-struction. Powerful computers now make it possible to produce more detailed models of the electrical, mechanical and thermal behavior of transformers than previously possible. The ever-increasing competition in the global market has put tremendous responsibilities on the transformer industry to increase transformer re-liability while reducing cost, since high quality, low cost products have become the key to survival. However, it is difficult, if at all possible, to meet today’s trans-former design demands via conventional design techniques.

Today, artificial intelligence is widely used in modeling nonlinear and large-scale systems, especially when explicit mathematical models are difficult to obtain or are completely lacking. Moreover, artificial intelligence is computationally ef-ficient in solving hard optimization problems.

The limitations of the analytical techniques as well as the progress of com-puters facilitated the development of numerical techniques for the solution of elec-tromagnetic field problems. Among the numerical techniques, the most popular method for the solution of electromagnetic field problems is the finite element method. A very real advantage of the finite element method is its ability to deal with complex geometries. Another advantage is that it yields stable and accurate solutions.

The subject of the book is Modern Transformer Design. This book introduces a novel approach to transformer design using artificial intelligence and numerical techniques.

The author worked in the transformer industry for 10 years before joining aca-demia. He has vast experience in the design, development and manufacturing of transformers. The author has developed the bulk of the results presented in the book during the last 10 years, while some of the results appear for the first time.

There is no other book including shell type transformer design by means of magnetic field analysis and artificial intelligence techniques. Most of the material in the book is an expanded and detailed version of the author’s original work in the field of transformer design. The basic philosophy of the book is that we learn by applying. That is why the book has many numerical examples that illustrate the use of the techniques for a variety of real-world transformer designs.

The book will be particularly useful to graduate and postgraduate students in electric power engineering devices, researchers in the design and implementation

x Preface

of power transformers, transformer designers and power engineering profession-als. More specifically:

1. Graduate and postgraduate students as well as researchers will learn new meth-odologies for transformer design optimization (TDO). Moreover, they will be able to apply and extend the methodologies of the book to the optimization of different types of transformers or to the optimization of other electrical ma-chines and devices. They will also find real and accurate data since all trans-former design examples are from actual constructed and tested transformers.

2. Transformer designers will be helped to apply artificial intelligence to optimiz-ing their transformer designs. In order to assist them, the book presents the ba-sic principles of artificial intelligence methods in separate chapters and in stand-alone form, i.e., the transformer designers will find the majority of the in-formation they need within the book. Moreover, transformer designers can ex-tend the methodologies of the book to optimize the designs of specific trans-former types and technologies they use at their transformer manufacturing plant.

3. Power engineering professionals working in electric utilities, industries, public authorities and design offices will find information to improve transformer specifications. They will find methodologies in the book that will help them in their transformer purchasing decisions. In particular, they will save money by purchasing the most cost-effective and energy-efficient transformers.

The material of the book is organized in three parts and eight chapters. Part I, which includes Chaps. 1 and 2, is devoted to the presentation of conventional transformer design. Part II, which includes Chapts. 3 to 5, presents the evaluation and optimization techniques that will be used in the third part of the book for the solution of a number of transformer design problems. Part III, which includes Chaps. 6 to 8, is dedicated to modern transformer design and it illustrates clearly how artificial intelligence and numerical techniques successfully solve a number of hard transformer design evaluation and optimization problems.

Chapter 1 is an introduction to transformer fundamentals. It describes the basic principles for the analysis of magnetic circuits, the correspondence between elec-tric and magnetic circuits, and the modeling of magnetic materials used in the con-struction of the transformer magnetic circuit. It presents a transformer equivalent circuit, a method to determine the parameters of the equivalent circuit, and formu-las to compute voltage regulation and efficiency. It defines the electrical character-istics of a transformer, e.g., rated power, rated voltages, frequency, no-load losses, load losses, and impedance voltage. It describes two interesting transformer oper-ating modes, i.e., overloading and parallel operation. It gives a list of standards that are typically used for transformer manufacturing. It presents the type, routine, and special tests that are performed on transformers. It classifies transformers ac-cording to their use, cooling medium, insulating medium, and core construction.

Preface xi

Finally, Chap. 1 describes the type and characteristics of transformers studied in this book.

Chapter 2 deals with the conventional design of wound core type transformers. It formulates the TDO problem and solves it using a multiple design method that is commonly referred to as the conventional TDO method. A design example of an actual commercial transformer is worked out throughout this chapter showing all the calculations that are needed to design a transformer. The example-driven pres-entation of the conventional TDO method makes this chapter unique in the trans-former design literature.

Transformers involve magnetostatic problems. These problems can be solved by analytical and numerical techniques. The limitations of the analytical tech-niques as well as the progress of computers has facilitated the development of numerical techniques. Among the numerical techniques, the most popular method in the solution of magnetostatic problems is the finite element method. A very strong advantage of the finite element method is its ability to deal with complex geometries. Another advantage is that it yields stable and accurate solutions. Chapter 3 presents the finite element method for the solution of linear and nonlin-ear magnetostatic problems, the latter being very common in transformer design. Carefully selected arithmetic examples make clear the application of the finite element method in the solution of linear and nonlinear magnetostatic problems.

Classification aims at predicting the future class, and forecasting aims at pre-dicting the future value of a system that is intrinsically uncertain. Chapter 4 briefly presents two artificial intelligence methods, namely decision trees and artificial neural networks. The decision tree methodology is a nonparametric inductive learning technique, able to produce classifiers for a given problem that can assess new, unseen situations and/or uncover the mechanisms driving this problem. The artificial neural network is a computer information processing system that is capa-ble of adequately representing nonlinear functions. The decision tree technique is appropriate for the solution of classification problems. The artificial neural net-work method is suitable for the solution of both classification and forecasting problems.

Chapter 5 is devoted to optimization and is organized into five sections. Section 5.1 is an introduction to optimization. Section 5.2 presents an active set method that effectively solves quadratic programming problems. Section 5.3 describes the sequential quadratic programming method, which is one of the best methods for solving nonlinearly constrained optimization problems. The sequential quadratic programming method iteratively solves a sequence of quadratic programming subproblems. Section 5.4 presents the branch-and-bound method, which, in con-junction with sequential quadratic programming, effectively solves mixed-integer nonlinear programming problems (such as the TDO problem of Chap. 7). Section 5.5 is devoted to the genetic algorithm method, which successfully solves complex optimization problems (such as the transformer no-load loss minimization prob-lem of Chap. 7). The four optimization methods that are presented in this chapter are accompanied by carefully selected and analytically solved arithmetic examples

xii Preface

that make clear the application of the methods to the solution of a variety of opti-mization problems.

Chapter 6 is devoted to the evaluation of transformer technical characteristics. Decision trees and artificial neural networks solve the no-load loss classification problem. Artificial neural networks solve the no-load loss prediction problem. Im-pedance voltage evaluation is implemented using a particular finite element model with detailed representation of winding geometry.

Chapter 7 deals with modern design optimization of wound core type trans-formers. Four methods are presented that solve important transformer design prob-lems. First, genetic algorithms are combined with artificial neural networks to op-timally group available individual cores into N transformers so as to minimize the total no-load loss of N transformers. This method significantly re-duces the no-load loss design margin as well as the cost of transformer main mate-rials. Second, decision trees and artificial neural networks successfully solve the winding material selection problem, thus avoiding the need to optimize the trans-former twice, once with copper and once with aluminum windings. Third, a mixed integer programming–finite element method is developed for solution of the TDO problem. Finally, a recursive genetic algorithm–finite element method is devel-oped to solve the TDO problem and is compared with the mixed integer pro-gramming–finite element method. The recursive genetic algorithm approach can also be very useful for the solution of other optimization problems in electric ma-chines and power systems.

4 N⋅

Chapter 8 deals with transformer selection by electric utilities and industrial transformer users. It reviews the classical total owning cost formula and it also in-troduces the external environmental cost due to transformer losses. Using the methodologies of this chapter, transformer users will save money by purchasing the most cost-effective and energy-efficient transformers.

Much of the material presented in this book was obtained through teamwork with colleagues at the National Technical University of Athens, the Technical University of Crete and Schneider Electric AE.

I would like to express my most sincere thanks to Professor Nikos Hatziargy-riou, supervisor of my PhD dissertation, for his continuous guidance, encourage-ment and support throughout my PhD and for introducing me to artificial intelli-gence based transformer design. Special thanks go to Professor Antonios Kladas for excellent and fruitful research collaboration in the area of numerical tech-niques for analysis of the transformer magnetic field.

I sincerely acknowledge the rich and ample experience gained while working in Schneider Electric AE and I am grateful to all my erstwhile senior colleagues. I would particularly like to express my sincere gratitude to Mr Athanasios Souflaris, Mr Yiannis Bakopoulos, Mr Spiros Elefsiniotis, Mr Dimitrios Paparigas, and Mr Dionissios Spiliopoulos for their support and guidance.

It was a great pleasure for me to collaborate with three PhD students in the area of transformer design. I would like to express my sincere thanks to Dr Marina Tsili, Dr Eleftherios Amoiralis and Dr Themistoklis Kefalas for our fruitful col-laboration.

Preface xiii I would like to thank Professor Nikola Rajakovic, Professor Vlastimir Glamo-

canin, Professor Suad Halilcevic, Professor Antonios Kladas and three more anonymous reviewers for the time invested to review this book and for their con-structive comments that helped me to improve the quality, presentation and or-ganization of the book.

Thanks are also due to Mr Anthony Doyle from Springer for his invitation to write this book and for believing in the project from the beginning as well as to Mr Simon Rees and Ms Claire Protherough from Springer who gave very good edito-rial input.

This book would not have been possible without the understanding and pa-tience of my wife Liza.

Chania, Greece Pavlos S. Georgilakis January 2009

Contents

Part I Conventional Transformer Design ....................................................... 1

1 Transformers .................................................................................................. 3 1.1 Introduction ............................................................................................. 3 1.2 Magnetic Circuits ..................................................................................... 4

1.2.1 General ......................................................................................... 4 1.2.2 Analysis of Magnetic Circuits ...................................................... 7 1.2.3 Flux Linkage ................................................................................ 9 1.2.4 Magnetic Materials ..................................................................... 10

1.3 Transformer Fundamentals .................................................................... 12 1.3.1 Equivalent Circuit ...................................................................... 12 1.3.2 Derivation of Equivalent Circuit Parameters .............................. 14 1.3.3 Voltage Regulation ..................................................................... 18 1.3.4 Efficiency ................................................................................... 23

1.4 Transformer Electrical Characteristics .................................................. 27 1.4.1 Rated Power ............................................................................... 27 1.4.2 Temperature Rise ....................................................................... 28 1.4.3 Ambient Temperature ................................................................. 28 1.4.4 Altitude of Installation ............................................................... 29 1.4.5 Impedance Voltage ..................................................................... 29 1.4.6 No-Load Losses .......................................................................... 29 1.4.7 Load Losses ................................................................................ 30 1.4.8 Rated Voltages ........................................................................... 31 1.4.9 Vector Group .............................................................................. 31 1.4.10 Frequency .................................................................................. 32 1.4.11 Noise ......................................................................................... 32 1.4.12 Short-Circuit Current ................................................................. 32 1.4.13 No-Load Current ....................................................................... 32

1.5 Transformer Operation .......................................................................... 33 1.5.1 Overloading ................................................................................ 33 1.5.2 Parallel Operation ....................................................................... 33 1.5.3 Load Distribution to Transformers in Parallel Operation ........... 34

1.6 Transformer Standards and Tolerances ................................................. 35 1.6.1 Transformer Standards ............................................................... 35 1.6.2 Tolerances .................................................................................. 36

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1.7 Transformer Tests ..................................................................................37 1.7.1 Type Tests ..................................................................................37 1.7.2 Routine Tests ..............................................................................37 1.7.3 Special Tests ...............................................................................39

1.8 Transformer Types .................................................................................39 1.8.1 Classification According to Transformer Use ............................40 1.8.2 Classification According to Transformer Cooling Method ........40 1.8.3 Classification According to Transformer Insulating Medium ....41 1.8.4 Classification According to Transformer Core Construction .....41

1.9 Transformers Studied in this Book ........................................................42 References .......................................................................................................43

2 Conventional Transformer Design ..............................................................45 2.1 Nomenclature .........................................................................................45 2.2 Introduction ............................................................................................49 2.3 Problem Formulation .............................................................................49

2.3.1 Objective Function .....................................................................50 2.3.2 Constraints ..................................................................................52 2.3.3 Mathematical Formulation of the TDO Problem ........................57 2.3.4 Characteristics of the TDO Problem ..........................................58

2.4 Conventional Transformer Design Optimization Method ......................59 2.4.1 Methodology ..............................................................................59 2.4.2 Case Study ..................................................................................62 2.4.3 Repetitive Transformer Design Process .....................................66

2.5 Example of Transformer Design Data ...................................................68 2.5.1 Values of Description Variables .................................................70 2.5.2 Values of Special Variables ........................................................70 2.5.3 Values of Default Variables .......................................................70 2.5.4 Values of Cost Variables ............................................................70 2.5.5 Values of Various Variables .......................................................71 2.5.6 Values of Conductor Cross-Section Calculation Variables ........71 2.5.7 Values of Design Variables ........................................................71

2.6 Calculation of Volts per Turn and Thickness of Core Leg ....................74 2.6.1 Calculation of Volts per Turn .....................................................74 2.6.2 Calculation of Thickness of Core Leg ........................................74 2.6.3 Example 2.1 ................................................................................76

2.7 Calculation of Layer Insulation .............................................................77 2.7.1 Layer Insulation of LV Winding ................................................78 2.7.2 Layer Insulation of HV Winding ................................................78 2.7.3 Example 2.2 ................................................................................78

2.8 Calculation of Winding and Core Dimensions ......................................79 2.8.1 Example 2.3 ................................................................................79

2.9 Calculation of Core Weight and No-Load Loss .....................................84 2.9.1 Example 2.4 ................................................................................86

Contents xvii 2.10 Calculation of Inductive Part of Impedance Voltage ............................. 87

2.10.1 Example 2.5 ................................................................................ 89 2.11 Calculation of Load Loss ....................................................................... 94

2.11.1 Example 2.6 ................................................................................ 94 2.12 Calculation of Impedance Voltage ......................................................... 99

2.12.1 Example 2.7 .............................................................................. 100 2.13 Calculation of Coil Length .................................................................. 100

2.13.1 Example 2.8 .............................................................................. 101 2.14 Calculation of Tank Dimensions ......................................................... 102

2.14.1 Example 2.9 .............................................................................. 102 2.15 Calculation of Winding Gradient and Oil Gradient ............................. 103

2.15.1 Example 2.10 ............................................................................ 103 2.16 Calculation of Heat Transfer ................................................................ 106

2.16.1 Example 2.11 ............................................................................ 108 2.17 Calculation of the Weight of Insulating Materials ............................... 110

2.17.1 Example 2.12 ............................................................................ 110 2.18 Calculation of the Weight of Ducts ..................................................... 114

2.18.1 Example 2.13 ............................................................................ 114 2.19 Calculation of the Weight of Oil .......................................................... 115

2.19.1 Example 2.14 ............................................................................ 115 2.20 Calculation of the Weight of Sheet Steel ............................................. 116

2.20.1 Example 2.15 ............................................................................ 117 2.21 Calculation of the Weight of Corrugated Panels ................................. 117

2.21.1 Example 2.16 ............................................................................ 117 2.22 Calculation of the Cost of Transformer Main Materials ...................... 117

2.22.1 Example 2.17 ............................................................................ 118 2.23 Calculation of Transformer Manufacturing Cost ................................. 119

2.23.1 Example 2.18 ............................................................................ 120 References .................................................................................................... 122

Part II Evaluation and Optimization Methods ........................................... 123

3 Numerical Analysis ..................................................................................... 125 3.1 Introduction ......................................................................................... 125

3.1.1 Magnetostatic Problems ........................................................... 125 3.1.2 Methods for the Solution of Magnetostatic Problems .............. 127

3.2 Finite Element Method ........................................................................ 128 3.2.1 Introduction .............................................................................. 128 3.2.2 Applications to Power Engineering .......................................... 129 3.2.3 Solution of Linear Magnetostatic Problems ............................. 130 3.2.4 Solution of Nonlinear Magnetostatic Problems ........................ 146

References .................................................................................................... 153

xviii Contents 4 Classification and Forecasting ...................................................................157

4.1 Introduction ..........................................................................................157 4.2 Automatic Learning .............................................................................158 4.3 Data Mining .........................................................................................158

4.3.1 Representation ..........................................................................159 4.3.2 Attribute Selection ....................................................................159 4.3.3 Model Selection ........................................................................159 4.3.4 Interpretation and Validation ....................................................159 4.3.5 Model Use ................................................................................160

4.4 Learning Set and Test Set ....................................................................160 4.4.1 Classification ............................................................................160 4.4.2 Forecasting ...............................................................................161

4.5 Decision Trees .....................................................................................162 4.5.1 Introduction ..............................................................................162 4.5.2 Applications to Power Systems ................................................163 4.5.3 General Characteristics .............................................................164 4.5.4 Top Down Induction .................................................................165 4.5.5 Optimal Splitting Rule ..............................................................167 4.5.6 Stop Splitting Rule ...................................................................170 4.5.7 Overview of Decision Tree Building Algorithm ......................173 4.5.8 Example 4.1 ..............................................................................174 4.5.9 Example 4.2 ..............................................................................179

4.6 Artificial Neural Networks ..................................................................185 4.6.1 Introduction ..............................................................................185 4.6.2 Applications to Power Systems ................................................186 4.6.3 ANN Types ...............................................................................187 4.6.4 Neuron Mathematical Model ....................................................188 4.6.5 ANN Architectures ...................................................................189 4.6.6 ANN Training ...........................................................................191 4.6.7 ANN Configuration ..................................................................205 4.6.8 Example 4.5 ..............................................................................207

4.7 Hybrid Decision Tree–Neural Network Classifier ...............................210 4.7.1 Example 4.6 ..............................................................................211

References .....................................................................................................212

5 Optimization ...............................................................................................219 5.1 Introduction ..........................................................................................219 5.2 Quadratic Programming .......................................................................222

5.2.1 Methodology ............................................................................222 5.2.2 Applications to Power Systems ................................................225 5.2.3 Example 5.1 ..............................................................................225

5.3 Sequential Quadratic Programming .....................................................231 5.3.1 Methodology ............................................................................231 5.3.2 Applications to Power Systems ................................................233

Contents xix

5.3.3 Example 5.2 .............................................................................. 233 5.4 Branch-and-Bound ............................................................................... 239

5.4.1 Methodology ............................................................................ 239 5.4.2 Applications to Power Systems ................................................ 241 5.4.3 Example 5.3 .............................................................................. 241

5.5 Genetic Algorithms .............................................................................. 244 5.5.1 Methodology ............................................................................ 244 5.5.2 Applications to Power Systems ................................................ 248 5.5.3 Example 5.4 .............................................................................. 249

References .................................................................................................... 256

Part III Modern Transformer Design ......................................................... 263

6 Evaluation of Transformer Technical Characteristics ............................ 265 6.1 Introduction ......................................................................................... 265 6.2 No-Load Loss Classification with Decision Trees and Artificial Neural Networks .................................................................................. 266

6.2.1 Introduction .............................................................................. 266 6.2.2 Individual Core ......................................................................... 267 6.2.3 Transformer .............................................................................. 281

6.3 No-Load Loss Forecasting with Artificial Neural Networks ............... 292 6.3.1 Introduction .............................................................................. 292 6.3.2 Forecasting Accuracy ............................................................... 294 6.3.3 Individual Core ......................................................................... 294 6.3.4 Transformer .............................................................................. 298

6.4 Impedance Voltage Evaluation with Numerical Models ..................... 301 6.4.1 Introduction .............................................................................. 301 6.4.2 Finite Element Model ............................................................... 302 6.4.3 Results and Discussion ............................................................. 317

References .................................................................................................... 325

7 Transformer Design Optimization ............................................................ 331 7.1 Introduction ......................................................................................... 331 7.2 No-Load Loss Reduction with Genetic Algorithms ............................. 332

7.2.1 Introduction .............................................................................. 332 7.2.2 Conventional Core Grouping Process ...................................... 332 7.2.3 Genetic Algorithm Solution to the TNLLR Problem ............... 334 7.2.4 Results ...................................................................................... 341

7.3 Winding Material Selection with Decision Trees and Artificial Neural Networks .................................................................................. 343

7.3.1 Introduction .............................................................................. 343 7.3.2 Creation of Knowledge Base .................................................... 344

xx Contents

7.3.3 Decision Trees ..........................................................................346 7.3.4 Adaptive Trained Neural Networks ..........................................349 7.3.5 Synthesis ...................................................................................359

7.4 Transformer Design Optimization with Branch-and-Bound ................359 7.4.1 Introduction ..............................................................................359 7.4.2 MIP-FEM Methodology ...........................................................360 7.4.3 Results and Discussion .............................................................364

7.5 Transformer Design Optimization with Genetic Algorithms ...............368 7.5.1 Introduction ..............................................................................368 7.5.2 Recursive GA-FEM Methodology ...........................................368 7.5.3 Results and Discussion .............................................................372

References .....................................................................................................374

8 Transformer Selection ................................................................................377 8.1 Introduction ..........................................................................................377 8.2 Total Owning Cost for Industrial and Commercial Users ....................378

8.2.1 Cost Evaluation Method ...........................................................378 8.2.2 Example 8.1 ..............................................................................382 8.2.3 Example 8.2 ..............................................................................385 8.2.4 Example 8.3 ..............................................................................385

8.3 Total Owning Cost for Electric Utilities ..............................................391 8.3.1 Cost Evaluation Method ...........................................................391 8.3.2 Example 8.4 ..............................................................................394 8.3.3 Example 8.5 ..............................................................................396

8.4 Proposed TOC Incorporating Environmental Cost ..............................400 8.4.1 Introduction ..............................................................................400 8.4.2 Cost Evaluation Method ...........................................................402 8.4.3 Example 8.6 ..............................................................................407 8.4.4 Example 8.7 ..............................................................................408 8.4.5 Example 8.8 ..............................................................................409 8.4.6 Example 8.9 ..............................................................................411 8.4.7 Example 8.10 ............................................................................417

References .....................................................................................................419

Index ..................................................................................................................423